Problem 14 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line passing through \((1,-1)\) and \((4,5)\)

Step-by-Step Solution

Answer

The standard form of the equation is \( 2x - y = 3 \).

Step 1: Find the Slope of the Line

The first step to find the equation of the line is to calculate its slope. The formula to find the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting the given points (1, -1) and (4, 5), we have \( m = \frac{5 - (-1)}{4 - 1} = \frac{6}{3} = 2 \). Therefore, the slope of the line is 2.

Step 2: Use Point-Slope Form to Find the Equation

We can use the point-slope form equation, which is \( y - y_1 = m(x - x_1) \), to find the equation of the line using one of the points and the slope. Using the point (1, -1) and the slope 2, the equation becomes \( y + 1 = 2(x - 1) \).

Step 3: Simplify the Equation to Slope-Intercept Form

Expand and simplify the equation \( y + 1 = 2(x - 1) \): \( y + 1 = 2x - 2 \). Subtract 1 from both sides to simplify further: \( y = 2x - 3 \). This is the slope-intercept form of the equation.

Step 4: Convert to Standard Form

The standard form of a linear equation is \( Ax + By = C \), where A, B, and C are integers, and A should be non-negative. Starting from the slope-intercept form \( y = 2x - 3 \), move all terms to one side: \( 2x - y = 3 \). This is already in standard form.

Key Concepts

Slope Calculation

Understanding how to calculate the slope is vital in deriving the equation of a line. The slope defines how steep the line is and in what direction it moves. It is denoted by the letter \( m \). Essentially, the slope is a measure of the "rise" over the "run," or in other words, how much the line goes up (or down) for a unit increase in the horizontal direction.

The formula for calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula tells us the difference in the \( y \)-values divided by the difference in the \( x \)-values, which gives the steepness and direction of the line. If the slope is positive, the line rises as it moves from left to right; if negative, it falls.

In the exercise, we used the points (1, -1) and (4, 5) to find the slope:

• Substitute: \( m = \frac{5 - (-1)}{4 - 1} = \frac{6}{3} = 2 \)

This means the line has a positive slope of 2, moving upwards right.

Point-Slope Form

The point-slope form of a linear equation is handy when you need to find the equation of a line given one point on the line and the slope. This form is expressed as:

• \( y - y_1 = m(x - x_1) \)

Where \( (x_1, y_1) \) is a known point, and \( m \) is the slope of the line.

Starting with this form allows you to quickly build the equation without needing both points, which simplifies things substantially, especially in real-world applications. For the problem at hand, using point \(1, -1)\) and slope \( m = 2 \), the point-slope form becomes:

• \( y + 1 = 2(x - 1) \)

This form clearly outlines the foundational elements of our specific line - where it comes from and where it's headed. It's perfect for converting to other forms like slope-intercept or standard form.

Standard Form of Linear Equation

The standard form of a linear equation is another way to express a line's equation using integer coefficients. It is given as:

• \( Ax + By = C \)

Where \( A \), \( B \), and \( C \) are integers, and \( A \) should ideally be non-negative. The beauty of this form is in how it presents the equation in a very structured manner, making it easy to analyze and work with, especially for those who are familiar with linear equations.

To transform a line equation from slope-intercept (or point-slope) form into standard form, rearrange it to ensure all terms are on one side, and simplify to meet the integer requirements. In our exercise, we converted:

• Starting from: \( y = 2x - 3 \)
• Rearrange to: \( 2x - y = 3 \)

Thus, \( 2x - y = 3 \) is the standard form, showcasing a clear and complete equation of the line first derived through slope calculations and point-slope manipulations.